6 Grain Boundary DiffusionEspecially for high symmetry boundaries, there is a very strong anisotropy of diffusion coefficients as a function of boundary type. This example is for Zn diffusing in a series of <110> symmetric tilts in copper.Note the low diffusion rates along low energy boundaries, especially 3.

7 Grain Boundary Sliding640°CGrain boundary sliding should be very structure dependent. Reasonable therefore that Biscondi’s results show that the rate at which boundaries slide is highly dependent on misorientation; in fact there is a threshold effect with no sliding below a certain misorientation at a given temperature.600°C500°CBiscondi, M. and C. Goux (1968). "Fluage intergranulaire de bicristaux orientés d'aluminium." Mémoires Scientifiques Revue de Métallurgie 55(2):

8 Grain Boundary Energy: DefinitionGrain boundary energy is defined as the excess free energy associated with the presence of a grain boundary, with the perfect lattice as the reference point.A thought experiment provides a means of quantifying GB energy, g. Take a patch of boundary with area A, and increase its area by dA. The grain boundary energy is the proportionality constant between the increment in total system energy and the increment in area. This we write: g = dG/dAThe physical reason for the existence of a (positive) GB energy is misfit between atoms across the boundary. The deviation of atom positions from the perfect lattice leads to a higher energy state. Wolf established that GB energy is correlated with excess volume in an interface. There is no simple method, however, for predicting the excess volume.

9 Measurement of GB EnergyWe need to be able to measure grain boundary energy.In general, we do not need to know the absolute value of the energy but only how it varies with boundary type, i.e. with the crystallographic nature of the boundary.For measurement of the anisotropy of the energy, then, we rely on local equilibrium at junctions between boundaries. This can be thought of as a force balance at the junctions.For not too extreme anisotropies, the junctions always occur as triple lines.

11 Zero-creep MethodThe zero-creep experiment primarily measures the surface energy.The surface energy tends to make a wire shrink so as to minimize its surface energy.An external force (the weight) tends to elongate the wire.Varying the weight can vary the extension rate from positive to negative, permitting the zero-creep point to be interpolated.Grain boundaries perpendicular to the wire axis counteract the surface tension effect by tending to decrease the wire diameter.

13 Definition of Dihedral AngleDihedral angle, c:= angle between the tangents to an adjacent pair of boundaries (unsigned). In a triple junction, the dihedral angle is assigned to the opposing boundary.1g1=g2=g323120°c1 : dihedral angle for g.b.1

14 Dihedral AnglesAn material with uniform grain boundary energy should have dihedral angles equal to 120°.Likely in real materials? No! Low angle boundaries (crystalline materials) always have a dislocation structure and therefore a monotonic increase in energy with misorientation angle (Read-Shockley model).The inset figure is taken from a paper in preparation by Prof. K. Barmak and shows the distribution of dihedral angles measured in a 0.1 µm thick film of Al, along with a calculated distribution based on an GB energy function from a similar film (with two different assumptions about the distribution of misorientations).

16 Unequal Energies, contd.A high g.b. energy on boundary 1 decreases the corresponding dihedral angle.Note that the dihedral angles depend on all the energies.1g1>g2=g332c1< 120°

17 WettingFor a large enough ratio, wetting can occur, i.e. replacement of one boundary by the other two at the TJ.g1>g2=g3 Balance vertical forces  g1 = 2g2cos(c1/2)Wetting g1  2 g2g11g2cosc1/2g3cosc1/232c1< 120°

22 Force BalanceConsider only interfacial energy: vector sum of the forces must be zero to satisfy equilibrium.These equations can be rearranged to give the Young equations (sine law):

23 Analysis of Thermal GroovesSurfaceCrystal 2WΨsCrystal 1d?β2WSurface 1Surface 2Grain BoundaryγGbγS2γS1It is often reasonable to assume a constant surface energy, gS, and examine the variation in GB energy, gGb, as it affects the thermal groove angles

24 Grain Boundary Energy Distribution is Affected by CompositionΔγ = 1.091 mΔγ = 0.46This anisotropy was confirmed by thermal groove measurements, which showed that the doped sample had a wider distribution of thermal groove widths and, therefore a wider distribution of energies. The diagram in lower right shows how we apply Young’s Eq, i.e. same local eq. at TJs to the balance between surfaces and GB energy. Image in upper right is an AFM of thermally grooved GBs.Ca solute increases the range of the gGB/gS ratio. The variation of the relative energy in undoped MgO is lower (narrower distribution) than in the case of doped material.76

25 Bi impurities in Ni have the opposite effectPure Ni, grain size: 20mmThe opposite trend was found in Bi-doped Ni. In this case, Bi suppressed the grain boundary anisotropy and a narrower range of energies was found.Very curious: Cu-Bi exhibits the opposite effect – the doped case is WIDER than the undoped. So, it is NOT possible to assume that segregation decreases or increases the anisotropy of GB energy.Bi-doped Ni, grain size: 21mmRange of gGB/gS (on log scale) is smaller for Bi-doped Ni than for pure Ni, indicating smaller anisotropy of gGB/gS. This correlates with the plane distribution77

27 Grain boundary energy: current status?Limited information available:Deep cusps exist for a few <110> CSL types in fcc (S3, S11), based on both experiments and simulation.Extensive simulation results [Wolf et al.] indicate that interfacial free volume is good predictor. No simple rules available, however, to predict free volume.Wetting results in iron [Takashima, Wynblatt] suggest that a broken bond approach (with free volume and twist angle) provides a reasonable 5-parameter model.If binding energy is neglected, an average of the surface energies is a good predictor of grain boundary energy in MgO [Saylor, Rohrer].Minimum dislocation density structures [Frank - see description in Sutton & Balluffi] provide a good model of g.b. energy in MgO, and may provide a good model of low angle grain boundary mobility.

28 Grain Boundary EnergyFirst categorization of boundary type is into low-angle versus high-angle boundaries. Typical value in cubic materials is 15° for the misorientation angle.Typical values of g.b. energies vary from J.m-2 for Al to 0.87 for Ni J.m-2 (related to bond strength, which is related to melting point).Read-Shockley model describes the energy variation with angle for low-angle boundaries successfully in many experimental cases, based on a dislocation structure.

29 Read-Shockley modelStart with a symmetric tilt boundary composed of a wall of infinitely straight, parallel edge dislocations (e.g. based on a 100, 111 or 110 rotation axis with the planes symmetrically disposed).Dislocation density (L-1) given by: 1/D = 2sin(q/2)/b  q/b for small angles.

30 Tilt boundarybDEach dislocation accommodates the mismatch between the two lattices; for a <112> or <111> misorientation axis in the boundary plane, only one type of dislocation (a single Burgers vector) is required.

31 Read-Shockley contd.For an infinite array of edge dislocations the long-range stress field depends on the spacing. Therefore given the dislocation density and the core energy of the dislocations, the energy of the wall (boundary) is estimated (r0 sets the core energy of the dislocation): ggb = E0 q(A0 - lnq), where E0 = µb/4π(1-n); A0 = 1 + ln(b/2πr0)

33 Read-Shockley contd.If the non-linear form for the dislocation spacing is used, we obtain a sine-law variation (Ucore= core energy): ggb = sin|q| {Ucore/b - µb2/4π(1-n) ln(sin|q|)}Note: this form of energy variation may also be applied to CSL-vicinal boundaries.

34 Energy of High Angle BoundariesNo universal theory exists to describe the energy of HAGBs.Based on a disordered atomic structure for general high angle boundaries, we expect that the g.b. energy should be at a maximum and approximately constant.Abundant experimental evidence for special boundaries at (a small number) of certain orientations for which the atomic fit is better than in general high angle g.b’s.Each special point (in misorientation space) expected to have a cusp in energy, similar to zero-boundary case but with non-zero energy at the bottom of the cusp.Atomistic simulations suggest that g.b. energy is (positively) correlated with free volume at the interface.

35 Exptl. vs. Computed Egb <100> Tilts <110> TiltsNote the presence of local minima in the <110> series, but not in the <100> series of tilt boundaries.S11 with (311) plane<110> TiltsS3, 111 plane: CoherentTwinHasson & Goux, Scripta metall

36 Surface Energies vs. Grain Boundary EnergyA recently revived, but still controversial idea, is that the grain boundary energy is largely determined by the energy of the two surfaces that make up the boundary (and that the twist angle is not significant).This is has been demonstrated to be highly accurate in the case of MgO, which is an ionic ceramic with a rock-salt structure. In this case, {100} has the lowest surface energy, so boundaries with a {100} plane are expected to be low energy.The next slide, taken from the PhD thesis work of David Saylor, shows a comparison of the g.b. energy computed as the average of the two surface energies, compared to the frequency of boundaries of the corresponding type. As predicted, the frequency is lowest for the highest energy boundaries, and vice versa.

39 Tilt versus Twist BoundariesIsolated/occluded grain (one grain enclosed within another) illustrates variation in boundary plane for constant misorientation. The normal is // misorientation axis for a twist boundary whereas for a tilt boundary, the normal is  to the misorientation axis. Many variations are possible for any given boundary.Misorientation axisTwist boundariesTilt boundariesgAgB

40 Separation of ∆g and nPlotting the boundary plane requires a full hemisphere which projects as a circle. Each projection describes the variation at fixed misorientation. Any (numerically) convenient discretization of misorientation and boundary plane space can be used.Distribution of normals for boundaries with S3 misorientation (commercial purity Al)Misorientation axis, e.g. 111, also the twist type location

41 Grain Boundary Distribution in MgO: [100] l(Dg, n)^[100]l(n|5°/[100])n^n^l(n|15°/[100])n^l(n|25°/[100])For any point on the plots, we have to define the BP normal in terms of its crystallographic directionThe problem is that, we can express it in terms of either of the two crystals, for example … blue = 100, red does not, we have to pick oneWe see that there are always peaks in the distribution for <100> BP normals, as it turns out that the other peaks occur when the BP normal is <100> in terms of the other crystal that we are not plottingSo, we start with one peak at each of the <100> poles when the <100> planes are reasonably close to each other, as the misorientation angle increases these planes move away from each other, and when the <100> planes of the two crystals are far enough from each other the peak splitsNot counting the low angle boundaries, every peak in the movie corresponds to a GB that has one of the two possible crystallographic BP normal directions is near <100>n^l(n|35°/[100])Every peak in l(Dg,n) is related to a boundary with a {100} plane

43 Grain boundary energy and populationFor all grain boundaries in MgO0.00.51.01.52.02.53.00.700.780.860.941.02ggb (a.u)ln(l+1)Population and Energy are inversely correlatedSaylor DM, Morawiec A, Rohrer GS. Distribution and Energies of Grain Boundaries as a Function of Five Degrees of Freedom. Journal of The American Ceramic Society 2002;85: Capillarity vector used to calculate the grain boundary energy distribution – see later slides.

47 Inclination DependenceInterfacial energy can depend on inclination, i.e. which crystallographic plane is involved.Example? The coherent twin boundary is obviously low energy as compared to the incoherent twin boundary (e.g. Cu, Ag). The misorientation (60° about <111>) is the same, so inclination is the only difference.

49 The torque termChange in inclination causes a change in its energy, tending to twist it (either back or forwards)df1

50 Inclination Dependence, contd.For local equilibrium at a TJ, what matters is the rate of change of energy with inclination, i.e. the torque on the boundary.Recall that the virtual displacement twists each boundary, i.e. changes its inclination.Re-express the force balance as (sg):torque termssurface tension terms

52 Torque effectsThe effect of inclination seems esoteric: should one be concerned about it?Yes! Twin boundaries are only one example where inclination has an obvious effect. Other types of grain boundary (to be explored later) also have low energies at unique misorientations.Torque effects can result in inequalities* instead of equalities for dihedral angles, if one of the boundaries is in a cusp, such as for the coherent twin.* B.L. Adams, et al. (1999). “Extracting Grain Boundary and Surface Energy from Measurement of Triple Junction Geometry.” Interface Science 7:

53 Aluminum foil, cross sectionsurfaceTorque term literally twists the boundary away from being perpendicular to the surfaceCross-section of a thin foil of Al.

57 Calculation of G.B. EnergyIn principle, one can measure many different triple junctions to characterize crystallography, dihedral angles and curvature.From these measurements one can extract the relative properties of the grain boundaries.The simpler procedure, described here, uses the dihedral angles and calculates the GB energy based on the 3 parameters of misorientation only, i.e. neglecting the torque term.The more complete calculation of GB energy is performed for all 5 macroscopic degrees of freedom. Since this does include the torque term, the capillarity vector can be used to accomplish this. The concept of the capillarity vector is described in subsequent slides.

66 Capillarity VectorThe capillarity vector is a convenient quantity to use in force balances at junctions of surfaces.It is derived from the variation in (excess free) energy of a surface.In effect, the capillarity vector combines both the surface tension (so-called) and the torque terms into a single vector quantity.The vector sum of the capillarity vectors of three boundaries joined at a triple line must (vector) sum to zero, or, more precisely, the vector sum cross the line tangent must be zero.It is therefore feasible to construct an algorithm that computes the anisotropy of grain boundary energy based on (iteratively) minimizing the error of the above vector sum at all triple lines.

67 Equilibrium at TJThe utility of the capillarity vector, x, can be illustrated by re-writing Herring’s equations as follows, where l123 is the triple line (tangent) vector. (x1 + x2 + x3) x l123 = 0Note that the cross product with the triple line tangent implies resolution of forces perpendicular to the triple line.Used by the MIMP group to calculate the GB energy function for MgO, based on a dataset with boundary normals (which imply dihedral angles) and grain orientations: Morawiec A. Method to calculate the grain boundary energy distribution over the space of macroscopic boundary parameters from the geometry of triple junctions. Acta mater. 2000;48:3525. Also, Saylor DM, Morawiec A, Rohrer GS. Distribution and Energies of Grain Boundaries as a Function of Five Degrees of Freedom. Journal of The American Ceramic Society 2002;85:3081.

70 Capillarity vector: componentsThe physical consequence of Eq (2) is that the component of x that is normal to the associated surface, xn, is equal to the surface energy, g.Can also define a tangential component of the vector, xt, that is parallel to the surface: where the tangent vector is associated with the maximum rate of change of energy.Sutton & Balluffi show how to derive the capillary pressure associated with a boundary from the capillarity vector. The result is the same as Herring’s original derivation and involves the “mean curvature”, k1+k2, (which is not the average curvature!), the mobility and the “interface stiffness”, which is the sum of the GB energy and its second derivative with respect to inclination. In this approach, principal curvatures must be evaluated, which is inconvenient for numerical calculations.

71 Computer Simulation of Grain GrowthSimulation of Grain Growth and G.B. populationsComputer Simulation of Grain GrowthFrom the PhD thesis project of Jason Gruber.MgO-like grain boundary properties were incorporated into a finite element model of grain growth, i.e. minima in energy for any boundary with a {100} plane on either side.Simulated grain growth leads to the development of a g.b. population that mimics the experimental observations very closely.The result demonstrates that it is reasonable to expect that an anisotropic GB energy will lead to a stable population of GB types (GBCD).

74 Population versus EnergyCorrelation of Grain Boundary Energy and PopulationSimulated data:Moving finite elements-3-2-11230.70.750.80.850.90.951.05ln(l)ggb (a.u.)(a)Experimental data: MgO-3-2-111.051.11.151.21.25ln(l)ggb (a.u.)(b)The relationship between the energy and the distribution the results from the simulation is also similar to the experiment.One of the most important issues we intend to investigate is how this scales with properties and how universal it is.Energy and population are strongly correlated in both experimental results and simulated results.Is there a universal relationship?

75 G.B. Energy: SummaryFor low angle boundaries, use the Read-Shockley model with a logarithmic dependence: well established both experimentally and theoretically.For high angle boundaries, use a constant value unless near a CSL structure with high fraction of coincident sites and plane suitable for good atomic fit.In ionic solids, a good approximation for the grain boundary energy is simply the average of the two surface energies (modified for low angle boundaries). This approach appears to be valid for metals also.In fcc metals, for example, low energy boundaries are associated with the presence of the close-packed 111 surface on one or both sides of the boundary.There are a few CSL types with special properties, e.g. high mobility sigma-7 boundaries in fcc metals.

76 Summary, contd.Although the CSL theory is a useful introduction to what makes certain boundaries have special properties, grain boundary energy appears to be more closely related to the energies of the two surfaces comprising the boundary. This holds over a wide range of substances.Grain boundary populations are inversely related to the associated energies.Grain boundary energies can be calculated from data on triple junctions that includes boundary normals and grain orientations.